Value versus Growth:
Time-Varying Expected Stock Returns
Huseyin Gulen∗
Krannert Graduate School of Management
Purdue University
Yuhang Xing†
Jones Graduate School of Management
Rice University
Lu Zhang‡
Stephen M. Ross School of Business
University of Michigan
and NBER
September 2008§
Abstract
Using the Markov switching framework of Perez-Quiros and Timmermann (2000), we show that
the expected value-minus-growth returns display strong countercyclical variations. Under a variety of flexibility proxies such as the ratio of fixed assets to total assets, the frequency of disinvestment, financial leverage, and operating leverage, we show that value firms are less flexible in
adjusting to worsening economic conditions than growth firms, and that inflexibility increases the
costs of equity in the cross section. The time-variations in the expected value premium highlight
the importance of conditioning information in understanding the cross section of average returns.
∗
Krannert Graduate School of Management, Purdue University, 403 West State Street, West Lafayette, IN 47907.
Tel: (765) 496-2689, fax: (765) 494-9658, and e-mail: hgulen@purdue.edu.
†
Jesse H. Jones Graduate School of Management, MS 531, Rice University, 6100 Main Street, Houston TX 77005.
Tel: (713) 348-4167, fax: (713) 348-5251, and e-mail: yxing@rice.edu.
‡
Finance Department, Stephen M. Ross School of Business, University of Michigan, 701 Tappan, ER 7605 Bus Ad,
Ann Arbor MI 48109-1234; and NBER. Tel: (734) 615-4854, fax: (734) 936-0282, and e-mail: zhanglu@bus.umich.edu.
§
We thank Murray Carlson (AFA discussant), Sreedhar Bharath, Amy Dittmar, Evan Dudley, Philip Joos,
Solomon Tadesse, Joanna Wu, and participants at the American Finance Association Annual Meetings for helpful
suggestions. This paper supersedes our working paper previously circulated under the title “Value versus growth:
Movements in economic fundamentals.” We are responsible for all the remaining errors.
1
Introduction
Value stocks earn higher average returns than growth stocks (e.g., Rosenberg, Reid, and Lanstein
1985, Fama and French 1992, 1993, and Lakonishok, Shleifer, and Vishny 1994). But the economic
sources of the value premium (the average return difference between value and growth stocks) have
been a source of heated debate. DeBondt and Thaler (1985), Lakonishok et al., and Daniel, Hirshleifer, and Subrahmanyam (1998) argue that the value premium is driven by investor overreaction.
The emerging investment-based asset pricing literature has responded by linking risk and expected
returns to firm characteristics and resurrected the view that fundamentals are important drivers
for cross-sectional expected returns (e.g., Berk, Green, and Naik 1999, Carlson, Fisher, and Giammarino 2004, and Zhang 2005). Although the underlying model structures are diverse, a central
prediction in this literature is that firms should differ in cyclical behavior of expected returns,
depending on their respective book-to-market ratios (see Section 2 for detailed explanations).
Consistent with theory, we find strong evidence of countercyclical movements in the expected
value-minus-growth return. Using the two-state Markov switching framework of Perez-Quiros and
Timmermann (2000), we find that in recessions, the expected excess returns of value stocks are most
strongly affected, and the expected excess returns of growth stocks are least affected, by worsening
aggregate economic conditions as measured by higher short-term interest rate and higher default
spread. By way of contrast, in expansions the expected excess returns of both value and growth
stocks have insignificant loadings on the short-term interest rate and the default spread. Because of
these asymmetries in response across the states of the economy, the expected value-minus-growth
return displays strong cyclical variations: It tends to spike upwards rapidly during recessions only
to decline more gradually in the ensuing expansions. We also document similar time-variations in
the conditional volatility and the conditional Sharpe ratio of the value-minus-growth strategy.
Guided by investment-based theories, we then examine the potential sources of the time-varying
expected value premium. Using a variety of proxies for real flexibility, we test the theoretical prediction that value firms are less flexible than growth firms in adjusting to worsening economic
2
conditions. The answer is strongly affirmative: We document that value firms have higher ratios of
fixed assets to total assets, higher frequency of disinvestment, higher financial leverage, and higher
operating leverage than growth firms. Moreover, using a composite inflexibility index that aggregates the information contained in the four aforementioned individual proxies, we show that firms
with less real flexibility incur higher costs of equity on average than firms with more real flexibility.
Our evidence echoes the conditional asset pricing literature on the importance of conditioning
information in understanding the cross section of average returns (e.g., Ferson and Harvey
1991, 1993, 1999, Jagannathan and Wang 1996, Campbell and Cochrane 1999, 2000, Lettau
and Ludvigson 2001, and Ang and Chen 2007).
The cross-sectional predictability literature
has traditionally emphasized unconditional models (e.g., Fama and French 1993, 1996).
But
unconditional models alone are unlikely to explain the time-variations in the expected value
premium because the evidence is all about the time series dynamics of the value premium. Lewellen
and Nagel (2006) argue that the conditional CAPM cannot explain the value premium because
the covariance between the value-minus-growth betas and the expected market risk premium is too
small. Our evidence is not inconsistent with theirs because we estimate the expected value premium
from a Markov switching predictive regressions framework, as opposed to the conditional CAPM.
Our Markov switching framework closely follows that of Perez-Quiros and Timmermann (2000).1
We differ in both economic question and the theoretical motivation. Perez-Quiros and Timmermann ask whether there exists a differential response in expected returns to negative monetary
policy shocks between small and large firms. Their study is motivated by the imperfect capital
markets theories (e.g., Bernanke and Gertler 1989, Gertler and Gilchrist 1994, and Kiyotaki and
Moore 1997). In contrast, we ask whether there exists a differential response in expected returns to
negative aggregate shocks between value and growth firms, an inquiry motivated by the investmentbased asset pricing theories (e.g., Carlson, Fisher, and Giammarino 2004 and Zhang 2005).
1
Similar regime switching models have been extensively used in empirical finance to successfully address diverse
issues including international asset allocation (e.g., Ang and Bekaert 2002a and Guidolin and Timmermann 2008b),
interest rate dynamics (e.g., Ang and Bekaert 2002b), capital markets integration (e.g., Bekaert and Harvey 1995),
and the joint distribution of stock and bond returns (e.g., Guidolin and Timmermann 2006).
3
Time-varying means and volatilities of the market excess returns have received much attention in empirical finance (e.g., Schwert 1989, Bekaert and Wu 2000, Ang and Bekaert 2007, and
Cochrane 2008). In contrast, there has been little direct evidence to date on the time-varying
expected returns of anomalies-based investment strategies in general, and on the time-varying expected value premium in particular. Guidolin and Timmermann (2008a) find evidence of four
regimes for the joint distribution of returns of the Fama-French (1993) three factors and emphasize
the resulting short-term market timing opportunities for investors. We instead focus on the strong
cyclical variations in the expected value premium and the underlying economic sources.
Our story proceeds as follows. Section 2 discusses the theoretical motivation for our empirical
analysis. Section 3 presents the estimation results from a univariate Markov switching model fitted
to the book-to-market deciles. Section 4 estimates a joint regime switching model for the value and
growth deciles simultaneously. Section 5 examines the sources of the time-variations in expected
returns related to the real flexibility of firms. Finally, Section 6 concludes.
2
Sources of the Time-Variations in the Expected Value Premium
Why should the expected returns of value firms covary more with recessions than the expected
returns of growth firms? Equivalently, why should the expected value premium display countercyclical variations? The investment-based asset pricing theories have recently provided some clues.
The key word is inflexibility. The basic idea is that due to a variety of sources, value firms are less
flexible than growth firms in adjusting to recessionary shocks. As a result, the risk and the expected
return of value firms are highest in recessions relative to those of growth firms. The theories have
suggested three distinct but related sources that can give rise to the relative inflexibility of value
firms: costly reversibility, operating leverage, and financial leverage.
Zhang (2005) argues that because of costly reversibility and time-varying price of risk, value
firms are less flexible than growth firms in scaling down to mitigate the impact of negative shocks.
As a result, value firms are riskier than growth firms in recessions when the price of risk is high.
4
Specifically, costly reversibility means that firms face higher costs in scraping than in expanding
the scale of productive assets (e.g., Ramey and Shapiro 2001). Because the assets of value firms
are less profitable than growth firms, value firms want to disinvest more in recessions. In contrast,
disinvesting is not as important for growth firms with more productive assets. Because disinvesting
is restricted by costly reversibility, the fundamentals of value firms are more adversely affected by
worsening economic conditions than the fundamentals of growth firms.
Time-varying price of risk reinforces the effect of costly reversibility. When the aggregate price
of risk is countercyclical (e.g., Campbell and Cochrane 1999), the discount rates of firms will be in
general higher in recessions than in expansions. Consequently, the expected net present values of
assets in place are even lower than they would be if the price of risk is constant, meaning that value
firms want to disinvest even more. Because scaling down is difficult, the fundamentals of value firms
are even more exposed to negative aggregate shocks than they would be with constant price of risk.
Because costly reversibility and time-varying price of risk deprive value firms of flexibility in
scaling down, value firms are riskier than growth firms in bad times. Combined with time-varying
price of risk, the countercyclical risk dynamics of the value-minus-growth strategy give rise to the
countercyclical variations in the expected value premium. Gala (2007) extends the economic mechanism of costly reversibility on risk and expected returns proposed by Zhang (2005) in the context
of an industry equilibrium model into a general equilibrium framework.2
The effect of costly reversibility on risk and expected returns are also studied by Kogan (2004)
and Cooper (2006). Kogan assumes that the rate of investment must be between zero and an upper
limit. The lower constraint is due to irreversibility, and the upper constraint is due to adjustment
costs. He finds that when the lower constraint is binding, the relation between book-to-market and
expected returns is positive. Using a dynamic investment model with fixed costs of investment,
2
Gomes, Kogan, and Zhang (2003) also use a general equilibrium model to link risk and expected return to firm
characteristics. However, in their model growth options are always riskier than assets in place. To generate positive
average value-minus-growth returns, Gomes et al. assume that growth options are distributed equally across firms.
Because more productive growth firms cannot invest more and grow faster, they pay more dividends and have shorter
equity duration than value firms. This prediction is counterfactual: Dechow, Sloan, and Soliman (2004) document that
long-duration stocks tend to be growth stocks, and that long-duration stocks have historically lower average returns.
5
Cooper finds that book-to-market is informative of the deviation of the actual capital stock from
the target level. This deviation in turn measures the sensitivity of stock returns to aggregate economic conditions. Although the underlying model structures are diverse, the central prediction is
largely in line with that from Zhang (2005) and Gala (2007).
Operating leverage also drives the time-variations in the expected value premium. Building on
Berk, Green, and Naik (1999), Carlson, Fisher, and Giammarino (2004) use a real options model to
study the dynamic behavior of monopolistic firms. When demand for the product of the firms in the
economy decreases, equity values fall relative to book values and revenues fall relative to the average
level. For value firms, equity values fall more relative to book values and revenues fall more relative
to the average, meaning that value firms should have higher operating leverage than growth firms.
Further, the fixed costs of production are fixed and do not decrease proportionally with revenues in
recessions, meaning that earnings (revenues minus variable and fixed costs) will decrease more than
proportionally relative to revenues. This operating leverage mechanism causes value firms to be
more affected by negative aggregate shocks than growth firms. As a result, the risk and expected
returns of value firms increase more than the risk and expected returns of growth firms in recessions.
The theories that we have discussed so far study firms that are all equity financed and ignore
the effect of financial leverage on risk and expected returns. Livdan, Sapriza, and Zhang (2008)
incorporate retained earnings, debt, costly external equity, and collateral constraints on debt capacity in the spirit of Hennessy and Whited (2005) into the investment-based asset pricing framework.
Livdan et al. find that leverage ratios increase risk and expected returns, and that this effect is
even more dramatic for the less profitable value firms than for the more profitable growth firms.
Livdan, Sapriza, and Zhang (2008) discuss two channels through which leverage affects risk and
expected returns. First, according to the standard leverage hypothesis in corporate finance, higher
leverage means that shareholders must bear a higher proportion of the asset risk and require a
higher risk premium to hold the stocks. Second, the inflexibility mechanism allows the underlying
asset risk to increase with leverage. Intuitively, more leveraged firms are burdened with more debt
6
and must pay more interests and retire a higher amount of the existing debt before financing new
investments. Thus, these firms are more likely to face binding collateral constraints, less flexible in
using investment to smooth dividends, riskier, and earn higher expected returns than less leveraged
firms. Finally, the leverage effect on risk and expected returns should be even more important
for the less profitable value firms than for the more profitable growth firms. The idea is that the
inflexibility due to high leverage ratios interacts with other sources of inflexibility such as costly
reversibility and operating leverage, both of which are more severe for less profitable firms.3
3
A Univariate Model of Time-Varying Expected Stock Returns
Section 3.1 describes the econometric framework that we use. Section 3.2 describes our data and
model specifications. Section 3.3 presents and discusses our estimation results.
3.1
The Econometric Framework
We adopt the Markov switching framework that explicitly allows for state dependence in expected
stock returns. To facilitate economic interpretation in lines of recessions and expansions, we allow
for only two possible states to be determined by the data.
Specifically, we use the Perez-Quiros and Timmermann (2000) Markov switching framework
with time-varying transition probabilities based on prior work of Hamilton (1989) and Gray (1996).
Let rt denote the excess return of a testing portfolio over period t and Xt−1 be a vector of conditioning variables. The Markov switching framework allows the intercept term, slope coefficients,
and volatility of excess returns to depend on a single, latent state variable, St :
rt = β 0,St + β ′St Xt−1 + ǫt ,
ǫ ∼ N (0, σ 2St )
(1)
in which N (0, σ 2St ) denotes a normal distribution with mean zero and variance, σ 2St . Two states,
St = 1 or St = 2, mean that the slopes and variance are either (β 0,1 , β ′1 , σ 21 ) or (β 0,2 , β ′2 , σ 22 ).
3
Obreja (2006) and Gomes and Schmid (2007) model defaultable debt and obtain similar predictions on the
leverage effect on risk and expected returns. Garlappi and Yan (2007) study an investment-based asset pricing model
with financial distress and show that financial leverage amplifies the magnitude of the value premium.
7
To specify how the underlying state evolves through time, we assume that the state transition
probabilities follow a first-order Markov chain:
pt = P (St = 1|St−1 = 1, Yt−1 ) = p(Yt−1 )
(2)
1 − pt = P (St = 2|St−1 = 1, Yt−1 ) = 1 − p(Yt−1 )
(3)
qt = P (St = 2|St−1 = 2, Yt−1 ) = q(Yt−1 )
(4)
1 − qt = P (St = 1|St−1 = 2, Yt−1 ) = 1 − q(Yt−1 )
(5)
in which Yt−1 is a vector of variables publicly known at time t − 1 and affect the state transition probabilities between time t − 1 and t. Prior studies have shown that the state transition
probabilities are time-varying and depend on prior conditioning information such as the economic
leading indicator (e.g., Filardo 1994 and Perez-Quiros and Timmermann 2000) or interest rates
(e.g., Gray 1996). Intuitively, investors are likely to possess superior information on the state
transition probabilities than that implied by the model with constant transition probabilities.
We estimate the parameters of the econometric model using maximum likelihood methods. Let
θ denote the vector of parameters entering the likelihood function for the data. Suppose the density
of the innovations, ǫt , conditional on being in state j, f (rt |St = j, Xt−1 ; θ), is Gaussian:
1
f (rt |Ωt−1 , St = j; θ) = p
exp
2πσ j
−(rt − β 0,j − β ′j Xt−1 )2
2σ j
!
(6)
for j = 1, 2. Ωt−1 denotes the information set that contains Xt−1 , rt−1 , Yt−1 , and lagged values of
these variables. The log-likelihood function is given by:
L(rt |Ωt−1 ; θ) =
T
X
log(φ(rt |Ωt−1 ; θ))
(7)
t=1
in which the density, φ(rt |Ωt−1 ; θ), is obtained by summing the probability-weighted state densities,
f (·), across the two possible states:
φ(rt |Ωt−1 ; θ) =
2
X
f (rt |Ωt−1 , St = j; θ)P (St = j|Ωt−1 ; θ)
j=1
8
(8)
and P (St = j|Ωt−1 ; θ) is the conditional probability of state j at time t given information at t − 1.
The conditional state probabilities can be obtained recursively:
P (St = i|Ωt−1 ; θ) =
2
X
P (St = i|St−1 = j, Ωt−1 ; θ)P (St−1 = j|Ωt−1 ; θ)
(9)
j=1
in which the conditional state probabilities, by Bayes’s rule, can be obtained as:
f (rt−1 |St−1 = j, Xt−1 , Yt−1 , Ωt−2 ; θ)P (St−1 = j|Xt−1 , Yt−1 , Ωt−2 ; θ)
P (St−1 = j|Ωt−1 ; θ) = P2
j=1 f (rt−1 |St−1 = j, Xt−1 , Yt−1 , Ωt−2 ; θ)P (St−1 = j|Xt−1 , Yt−1 , Ωt−2 ; θ)
(10)
Following Gray (1996) and Perez-Quiros and Timmermann (2000), we iterate on equations (9) and
(10) to derive the state probabilities, P (St = j|Ωt−1 ; θ), and obtain the parameter estimates of the
likelihood function. Variations in the state probabilities can be interpreted as evidence supporting
the notion of time-variations in expected stock returns.
3.2
Data and Model Specifications
We use as testing assets the excess returns of the decile portfolios formed on book-to-market equity.
Excess returns are defined in excess of the one-month Treasury bill rate. The data for the decile
returns and Treasury bill rates are from Kenneth French’s Web site. The sample period is from
January 1954 to December 2007 with a total of 648 monthly observations. Following Perez-Quiros
and Timmermann (2000), we start the sample from January 1954 to conform with the period after
the Treasury-Federal Reserve Accord that allows the Treasury bill rates to vary freely.
Table 1 reports the moments of monthly excess returns for ten book-to-market deciles. The
mean excess return increases virtually monotonically from 0.48% per month for the growth decile to
0.97% for the value decile. The value-minus-growth portfolio earns an average return of 0.48% with
a volatility of 4.28% per month, meaning the average return is more than 2.8 standard errors from
zero. Most of the deciles are negatively skewed, but the value-minus-growth portfolio is positively
skewed. Finally, all portfolios are fat-tailed with kurtosis coefficients larger than three.
To provide some initial sense on the time-variations in the expected value premium, Figure 1 pro-
9
vides a bar graph for the annual value-weighted value-minus-growth returns. The value portfolio is
the decile with the highest book-to-market equity, and the growth portfolio is the decile with lowest
book-to-market equity. Figure 1 suggests countercyclical variations in the expected value premium.
For example, the realized value-minus-growth return hits the bottom during the period immediately
before or shortly after the beginning of the recession in 2000. However, the value decile then outperforms the growth decile dramatically in earlier 2000s. This pattern of value first sinking below and
then rising above growth also can be observed in the recessions of 1957–58, 1981–82, and 1990–91.
Although outside our sample period, untabulated results also show that this dynamic profitability
pattern of the value-minus-growth strategy appears in the Great Depression. Interestingly, the value
portfolio underperforms the growth portfolio by 13.8% per annum in 2007, suggesting that the U.S.
economy is heading towards or is already in a recession at the time of this writing in September 2008.
To quantify the importance of the cyclical variations, we model the excess returns for each of
the book-to-market portfolios as a function of an intercept term and lagged values of the one-month
Treasury bill rate, the default spread, changes in the money stock, and the dividend yield. All the
variables are common predictors of stock market excess returns. We use the one-month Treasury
bill rate, T B, as a state variable to proxy for the unobserved expectations of investors on future
economic activity. A negative shock to real economic growth might lead to an increase in the
current and expected inflation, which in turn induces a higher Treasury bill rate. Thus, interest
rates and stock returns are likely to be negatively correlated contemporaneously (e.g., Fama and
Schwert 1977, Fama 1981, Campbell 1987, Glosten, Jagannathan, and Runkle 1993). Also, the
Federal Reserve typically raises short-term interest rates in expansions to curb inflation pressures
and lowers short-term interest rates in recessions to stimulate economic growth.
The default spread, DEF , is the difference between yields on Baa- and Aaa-rated corporate
bonds from Ibbotson Associates. Value firms are likely to be more exposed to bankruptcy risks
during recessions than growth firms, meaning that the returns of value stocks should have higher
loadings on the default spread than the returns of growth stocks. Further, the empirical macroe-
10
conomics literature shows that the default spread is one of the strongest forecasters of the business
cycle (e.g., Stock and Watson 1989 and Bernanke 1990). Not surprisingly, the default spread has
been used as a primary conditioning variable in the literature on stock market predictability (e.g.,
Keim and Stambaugh 1986, Campbell 1987, Fama and French 1989, and Kandel and Stambaugh
1990). Indeed, Jagannathan and Wang (1996) use the default spread as the only instrument in
modeling the expected market risk premium in their influential study of the conditional CAPM.
The growth in the money stock, ∆M , is defined as the 12-month log difference in the monetary
base reported by the Federal Reserve Bank in St. Louis. We use the change in the money supply
to measure the liquidity changes in the economy. ∆M also measures monetary policy shocks that
can affect aggregate economic conditions. The dividend yield, DIV , is defined as dividends on the
value-weighted CRSP market portfolio over the previous 12 months divided by the stock price at
the end of the month. The variable has been shown to predict future stock returns (e.g., Campbell
and Shiller 1988). The dividend yield captures mean reversion in expected stock returns because
a high dividend yield means that dividends are discounted as a higher rate.
For each book-to-market decile portfolio, indexed by i, we estimate the following model:
rti = β i0,St + β i1,St T Bt−1 + β i2,St DEFt−1 + β i3,St ∆Mt−2 + β i4,St DIVt−1 + ǫit
(11)
in which rti is the monthly excess return for the ith decile portfolio formed on book-to-market equity, ǫit ∼ N (0, σ 2i,St ), and St = {1, 2}. Following Perez-Quiros and Timmermann (2000), we lag
the one-month Treasury bill rate, the default spread, and the dividend yield by one period, and we
lag the growth in money supply by two periods to allow for the publication delay for this variable.
The conditional variance of excess returns, σ 2i,St is allowed to depend on the state of the economy:
log(σ 2i,St ) = λiSt
(12)
For simplicity, we do not include ARCH terms or instrumental variables in the volatility equation.
11
State transition probabilities are specified as follows:
i
pit = P (Sti = 1|St−1
= 1, Yt−1 ) = Φ(π i0 + π i1 T Bt−1 )
i
1 − pit = P (Sti = 2|St−1
= 1)
i
qti = P (Sti = 2|St−1
= 2, Yt−1 ) = Φ(π i0 + π i2 T Bt−1 )
i
1 − qti = P (Sti = 1|St−1
= 2)
(13)
(14)
(15)
(16)
in which Sti is the state indictor for the ith portfolio and Φ is the cumulative density function of a
standard normal variable. Following Gray (1996), we capture the information of investors on state
transition probabilities parsimoniously through the use of the one-month Treasury bill rate.
3.3
Estimation Results
We now present the estimation results for the ten book-to-market deciles.
The Interpretation of the States
Table 2 shows that state 1 is associated with low average returns and high conditional volatilities and that state 2 is associated with high average returns and low conditional volatilities. This
evidence means that state 1 is the recession state and state 2 is the expansion state.
Specifically, moving from the growth decile to the value decile, we observe that the constant
term is universally lower in state 1 and higher in state two. The constant term in state 1 also
is estimated with more precision than in state 2. Eight out of ten constant terms in state 1 are
significant at the five percent level and all ten estimates are negative. In contrast, only one out of
ten constant terms in state 2 is significant. Further, the difference in the constant term between
states 1 and 2 increases in magnitude from 0.80% per month for the growth portfolio to 9.2% per
month for the value portfolio. This evidence offers the first indication that the expected return of
the value portfolio reacts more to negative aggregate shocks than that of the growth portfolio.
Table 2 also shows that all the volatilities are estimated quite precisely with small standard
12
errors. Further, the volatility estimates are countercyclical: All ten deciles have volatilities in the
recession state that are about twice as large as those in the expansion state. This pattern confirms
the findings of Schwert (1989) that stock return volatilities are higher in recessions than those in
expansions. However, the difference in volatilities between the two states is largely similar in magnitude across the ten deciles. This evidence contrasts with that in Perez-Quiros and Timmermann
(2000) that the volatilities of small firms are more strongly affected by the recession state.
We can further clarify the interpretation of the two hidden states. Figure 2 plots the conditional transition probabilities of being in state 1 at time t conditional on the information set at
time t − 1, P (St = 1|Ωt−1 ; θ), for the value and growth portfolios. We also overlay the transition
probabilities with historical NBER recession dates. In our specification, the conditional transition
probabilities depend on lagged conditioning information and reflect the perception of investors on
the conditional likelihood of being in state 1 in the next period. From Figure 2, the transitional
probabilities of being in state 1 are all quite high during the eight recessions since World War
II. State 1 also captures times of high volatility such as October 1987 that is not in a recession.
However, the correlation between the state probabilities for the value portfolio with the procyclical
one-month Treasury bill rate is −0.20. And the correlation between the state probabilities for the
growth portfolio and the countercylical default spread is 0.49. On balance, the evidence suggests
that the state probabilities are countercyclical, meaning that state 1 with low average returns and
high volatilities can largely be interpreted as the recession state.
Conditional Mean Equations
The focus of our analysis is on the conditional mean equations. Table 2 shows that coefficients
on the one-month Treasury bill rate are all negative for the ten book-to-market deciles in state 1.
All the coefficients are significant at the five percent level. More important, the magnitude of the
coefficients varies systematically with book-to-market. Moving from the growth decile to the value
decile, the coefficients increase in magnitude virtually monotonically from −5.68 with a standard
error of 1.54 to −11.67 with a standard error of 3.28. This evidence means that in recessions, value
13
firms are more affected by interest rate shocks than growth firms.
By way of contrast, in the expansion state the excess returns of the book-to-market portfolios
are not much affected by the short-term interest rates. Although all the coefficients on the Treasury
bill rate are still negative, only three out of ten are significant. In particular, the coefficient for
the growth portfolio is −1.45, which is even slightly higher in magnitude than that for the value
portfolio, −1.34. However, both coefficients are within 1.2 standard errors of zero.
Table 2 also reports systematical variations in the coefficients of the portfolio excess returns on
the default spread. In the recession state, all the deciles generate coefficients of the default spread
that are positive and significant at the five percent level. More important, moving from the growth
portfolio to the value portfolio, the coefficient increases largely monotonically from 4.02 with a
standard error of 0.88 to 7.31 with a standard error of 1.79. But in the expansion state, none of the
ten estimated coefficients on the default spread are significant, although nine of ten remain positive.
There is some, albeit weak, evidence that growth firms respond more to the default premium than
value firms in the expansion state. The coefficient of growth firms in state 2 is 1.77 with a standard
error of 0.96, the coefficient that of value firms is only 0.38 with a standard error of 0.62. On
balance, however, the evidence suggests that the default spread mainly affect the expected returns
in the recession state and particularly for value firms.
The coefficients on the growth in money supply are not significant in our specification. These coefficients are all positive in recessions, meaning that higher monetary growth is related with higher
expected returns. A possible explanation is that the Federal Reserve increases the money supply in
recessions, during which the expected excess returns of the testing portfolios are higher. Turning to
the coefficients on the dividend yield, we observe that in the recession state, the coefficient for the
growth portfolio is positive but insignificant, 0.22 with a standard error of 0.34. In contrast, the coefficient for the value portfolio in the recession state is 1.52, which is more than two standard errors
from zero. However, six out of ten book-to-market deciles have insignificant coefficients on the dividend yield including the ninth decile. In the expansion state the growth portfolio has a significant
14
coefficient of 0.83 (standard error = 0.28), but the remaining deciles have insignificant coefficients.
Our results so far are consistent with the flexibility hypothesis that value firms are more affected
by worsening economic conditions than growth firms. However, they do not test whether the
differential responses between value and growth firms are statistically significant. We now report a
set of likelihood ratio tests for the existence of two states in the conditional mean equation, again
following Perez-Quiros and Timmermann (2000). We condition on the existence of two states in the
conditional volatility. This step is necessary because as pointed out by Hansen (1992), the standard
likelihood ratio test for multiple states is not defined because the transition probability parameters
are not identified under the null of a single state. The resulting likelihood ratio statistic follows a
standard chi-squared distribution. More formally, we test the null hypothesis that the coefficients
on the one-month Treasury bill rate, the default spread, the growth rate of money supply, and the
dividend yield are equal across states, i.e., β ik,St =1 = β ik,St =2 , k = 1, 2, 3, 4 for each testing portfolio i.
Table 3 shows that the state dependence in the conditional mean equations is indeed statistically significant. The p-values for the likelihood ratio tests are equal or smaller than one percent
for seven out of ten deciles, meaning that the null hypothesis is strongly rejected. In particular,
the null hypothesis is rejected at the one percent level for the value and growth deciles.
4
A Joint Model of Expected Value and Growth Returns
We generalize the previous framework by estimating a bivariate Markov switching model for the
excess returns on the value and the growth portfolios. Relative to the univariate framework that
we have estimated separately for each portfolio, the bivariate framework offers several advantages.
First, the joint framework allows us to impose the condition that the recession state occurs simultaneously for both value and growth portfolios. In so doing, we can obtain more precise estimates
of the underlying state. The joint model also provides a natural framework for modeling the timevarying expected value premium. Finally, the joint model allows us to formally test the hypothesis
that value firms display stronger cyclical variations in the expected returns than growth firms.
15
4.1
Model Specifications
Let rt = (rtg , rtv )′ be the (2 × 1) vector consisting of the excess returns on the growth portfolio, rtg ,
and those on the value portfolio, rtv . We specify the bivariate Markov switching model as follows:
rt = β 0,St + β 1,St T Bt−1 + β 2,St DEFt−1 + β 3,St ∆Mt−2 + β 4,St DIVt−1 + ǫt
in which β k,St is a (2 × 1) vector with the elements
N (0, ΣSt ), St = {1, 2}, is a vector of residuals. ΣSt
β gk,St
!
for k = 1, 2, 3, 4 and ǫt ∼
β vk,St
is a positive semi-definite (2 × 2) matrix
that contains the variances and covariances of the residuals of the excess returns of the value and
growth portfolios in state St . The diagonal elements of this variance-covariance matrix, Σii,St , take
the similar form as in the univariate model: log(Σii,St ) = λiSt . And the off-diagonal elements,
Σij,St , assume a state-dependent correlation between the residuals, denoted ρSt . More precisely,
Σij,St = ρSt (Σii,St )1/2 (Σjj,St )1/2 for i 6= j. We maintain the transition probabilities from the
univariate model, but now the same state drives both the value and the growth portfolios.
4.2
Estimation Results
Table 4 presents the results from the bivariate Markov switching model. Most important, the pattern of differential coefficients on the interest rates and on the default spreads in the conditional
mean equations across the value and growth deciles is quantitatively similar to that from the univariate specifications. Moving from the growth decile to the value decile, the coefficient on the Treasury
bill rate increases in magnitude from −6.74 (standard error = 2.18) to −10.76 (standard error =
2.25) in the recession state. And the coefficient on the default spread increases from 4.60 (standard
error = 1.38) for the growth portfolio to 7.76 (standard error = 1.29) for the value portfolio.
We also present the likelihood ratio tests on the hypothesis that the difference across states 1
and 2 in the coefficient of the value portfolio exceeds that in the coefficient of the growth portfolio.
Formally, for each set of coefficients indexed by k, we test the null hypothesis that:
β gk,1 − β gk,2 = β vk,1 − β vk,2
16
(17)
against the alternative hypothesis that the difference in coefficients is larger for the value portfolio.
The null is strongly rejected at the five percent significance level.
Imposing the same state across the value and growth deciles changes several results from the
univariate specifications. The constant term continues to be lower in state 1 than that in state 2,
but it no longer increases in magnitude from the growth decile to the value decile. Indeed, the null
hypothesis that the degree of asymmetry across the states is identical across the value and growth
portfolios cannot be rejected with a p-value of 0.30. Further, although the likelihood ratio test rejects the null hypothesis given by equation (17) for the coefficients on the growth in money supply
and on the dividend yield, none of the estimates are individually significant. Thus, the variations
in the coefficients on the dividend yield across the book-to-market deciles in the univariate specifications in Table 2 do not survive the restriction of a single latent state across the testing portfolios.
4.3
Cyclical Variations in the Expected Excess Returns, the Conditional
Volatilities, and the Conditional Sharpe Ratios
Figure 3 plots the expected excess returns for the value portfolio and the growth portfolio in Panel
A and B, respectively, as well as for the value-minus-growth portfolio in Panel C. The solid lines use
the estimates from the bivariate Markov switching model, and the dashed lines use the estimates
from the univariate model. From Panels A and B, the series from the univariate and the bivariate
models are largely similar. The expected excess returns of both value and growth firms tend to
increase rapidly during recessions and decline more gradually during expansions.
Panel C reports some discrepancy in the expected value-minus-growth returns estimated from
the univariate and the bivariate models. To the extent that the two expected return series differ,
we rely more on the series from the bivariate model to draw our inferences. The reason is simple:
The underlying state is designed to capture shocks to aggregate economic conditions and it makes
sense to impose the restriction that the state applies to value and growth firms simultaneously. Estimating the Markov switching model separately for the individual portfolios, while an informative
first step, is likely to contaminate the latent states with portfolio-specific shocks.
17
The solid line in Panel C of Figure 3 shows that the expected value-minus-growth returns are
positive for 472 out of 648 months, about 73% of the time. The mean is 0.39% per month, which
is more than 14 standard errors from zero. More important, the expected value premium displays
time-variations closely related to the states of the economy: The series tends to be small and even
negative prior to and during the early phase of recessions but to increase sharply during later stages
of recessions. The evidence is consistent with the theories discussed in Section 2: As a recession
deepens, value firms quickly lose flexibility in adjusting to adverse economic conditions and their
assets become riskier, causing investors to require a higher risk premium for holding their stocks.
Time-variations in expected returns can be driven by the variations in conditional volatilities,
the variations in conditional Sharpe ratios, or both. Panel A of Figure 4 plots the conditional
volatilities for the value and growth portfolios. The volatilities reflect the switching probabilities
and not just the volatilities of returns in a given state. Panel A shows that the conditional volatilities tend to spike upwards during most recessions for both value and growth firms. Value firms also
have slightly higher volatilities than growth firms: The average ratio of the conditional volatility
of the value portfolio over the conditional volatility of the growth portfolio is 1.06 in our sample.
Panel B of Figure 4 plots the estimates of the conditional Sharpe ratios for value and growth
firms from the bivariate Markov switching model. The Sharpe ratio dynamics are similar for the
value and growth portfolios and both display strong cyclical variations. The Sharpe ratios tend to
increase rapidly during recessions and to decline more gradually in expansions. Thus, the cyclical
variations in expected excess returns for value and growth firms reported in Panel A of Figure 3 are
driven by similar variations in both conditional volatilities and conditional Sharpe ratios. Panel B
of Figure 4 also shows that the value decile has mostly higher conditional Sharpe ratios than the
growth decile, especially in the early 2000s. Over the entire sample, the mean conditional Sharpe
ratio is higher for value firms than for growth firms: 0.66 vs. 0.38 per annum.
To further illustrate the difference in Sharpe ratio between the value and growth deciles, Figure
5 plots the expected return, conditional volatility, and conditional Sharpe ratio for the value-minus18
growth portfolio. Because volatilities and Sharpe ratios are not additive, these time-varying moments are estimated from using the value-minus-growth returns in the univariate Markov switching
model. From Panel A, the expected value premium is closer to that estimated from the bivariate
model, especially in the 1990s, than to the difference between expected value and growth returns
that are separately estimated from the univariate model. In particular, the expected value premium
is positive for 557 out of 648 months, or 86% of the time. The mean of the expected return is 0.52%
per month and is more than 20 standard errors from zero.
From Panels B and C of Figure 5, the conditional volatilities and the conditional Sharpe ratios
of the value-minus-growth portfolio both display strong countercyclical variations. In particular,
the Sharpe ratios tend to spike up during recessions only to decline gradually in the subsequent
expansions. The mean of the conditional Sharpe ratios in our sample is 0.15 per annum.
5
Are Value Firms Less Flexible Than Growth Firms?
We have shown that the expected excess returns of value firms display stronger countercyclical
variations than the expected excess returns of growth firms. The evidence is largely consistent with
the investment-based asset pricing theories discussed in Section 2. Guided by these theories, we
now examine the potential sources of the time-variations in the expected value premium. Specifically, we test the hypothesis that value firms are less flexible than growth firms in adjusting to
negative aggregate shocks. To this end, we perform descriptive tests using a variety of measures
of flexibility in the context of the book-to-market deciles. We also use the Fama-MacBeth (1973)
cross-sectional regressions and the Fama-French (1993) portfolio approach to test if flexibility is
related to the costs of equity capital in the cross section.
Our sample is from the merged CRSP and Compustat database from 1972 to 2007. The starting period is determined by data availability of some variables required in constructing flexibility
proxies. We use all NYSE, Amex, and Nasdaq firms excluding financial firms with four-digit SIC
codes between 6000 and 6999 and utilities with four-digit SIC codes between 4900 and 4999. CRSP
19
contains monthly prices, shares outstanding, dividends, and returns for publicly traded firms. Compustat annual research files provide the relevant accounting information.
5.1
Measures of Real Flexibility
Direct measures of real flexibility are not available. As such, we propose several variables that aim
to proxy for the flexibility. The first proxy is the ratio of fixed assets over total assets, denoted
F A/T A. We use gross property, plant, and equipment (Compustat annual item 7) to measure fixed
assets, and use item 6 to measure total assets. As discussed by Rajan and Zingales (1995), F A/T A
is related to asset specificity and tangibility. Because costly reversibility primarily applies to investment on property, plant, and equipment, a higher F A/T A means less real flexibility. Saunders,
Strock, and Travlos (1990) provide a different reason for using F A/T A as a proxy for flexibility:
Fixed costs of production are proportional to the scale of fixed assets, meaning that F A/T A also
is a measure of operating leverage.
The second proxy for real flexibility is the frequency of disinvestment. We measure this frequency as the ratio of the number of firms that have disinvested in at least one year during the
past three years divided by the total number of firms at the current year for a given portfolio. A
firm is considered as having disinvested if its sales of property, plant, and equipment (Compustat
annual item 107) is positive. The frequency of disinvestment measures the degree with which a firm
is susceptible to costly reversibility. The idea is that the more frequently a firm needs to disinvest,
the more susceptible the firm will be to costly reversibility, and less flexible the firm will be. Note
that the frequency of disinvestment is only defined for a given portfolio. We also use a firm-specific
measure of disinvestment, which is a dummy variable that takes the value of one if the firm has
disinvested for at least one year during the previous three years and takes the value of zero otherwise.
The third proxy for real flexibility is operating leverage. Carlson, Fisher, and Giammarino
(2004) argue that operating leverage is an important driving force of the value premium. The idea
is that if the fixed costs are proportional to the book value, value firms will have higher operating
20
leverage, which in turn gives rise to higher risk than growth firms. We measure operating leverage
as the ratio of the percentage change in operating income before depreciation (Compustat annual
item 13) to the percentage change in sales (item 12). However, this ratio can be negative in a given
year because sales and earnings can temporarily move in opposite directions. Further, if the change
in sales is too small in a given year, the ratio will become unrealistically high. To obtain a sensible
estimate of operating leverage for a given firm, we take the three-year moving average of the ratio
of the percentage change in operating income before depreciation to the percentage change in sales
as the measure of operating leverage. Moreover, if the ratio is negative for every year during the
recent three years, we set its value to be missing in the current year.
The final individual proxy we use for real flexibility is financial leverage. The idea is similar to
that of operating leverage. Interest expenditures add to the amount of fixed costs that firms incur
to keep in production. The earnings of firms with high fixed costs will vary more with economic
downturns than the earnings of firms with low fixed costs. Moreover, using a dynamic asset pricing
model with debt, Livdan, Sapriza, and Zhang (2008) argue that more leveraged firms are burdened
with more debt and must repay existing debt before financing new investments. As such, more
leveraged firms are more likely to be constrained financially, are less flexible, and are riskier in assets and equity values. Following Fama and French (1992), we measure financial leverage, denoted
A/M E, as the ratio of the book value of assets (Compustat annual item 6) over the market value
of equity. For portfolios formed in June of calendar year t, both book assets and market equity
are measured in fiscal year ending in calendar year t − 1. Using the debt-to-equity ratio (the ratio
of total assets minus the book equity over the market equity) to measure financial leverage as in
Bhandari (1988) yields largely similar results (not reported).
Each of these four flexibility proxies has its own merits but none of them are likely to be perfect.
To minimize the effects of their individual measurement errors and to maximize their combined
information content, we construct a composite inflexibility index by aggregating over these variables. Specifically, we sort firms into deciles in June of each year t on financial leverage, operating
21
leverage, and the fixed-to-total assets ratio and record the decile assignments. For disinvestment,
we assign a value of ten to the firms that have disinvested during the previous three years and a
value of one to the firms that have not disinvested. We then take the average of the decile indexes
for a given firm to obtain the composite inflexibility index assigned to the firm in the current year
t. Higher value of the inflexibility index means less flexibility.
5.2
Empirical Results
The Variations of Real Flexibility Across the Book-to-Market Portfolios
We first compute the flexibility measures for each firm and define the portfolio-level measures as the
cross-sectional averages of the proxies across firms in each book-to-market decile. Table 5 reports
the time series averages over the period from 1972 to 2007 for all the portfolio-level flexibility proxies.
Moving from the growth decile to the value decile, we observe a virtually monotonically increasing relation between book-to-market equity and the measures of inflexibility. The fixed-to-total
assets ratio increases from 0.36 for growth firms to 0.55 for value firms, and the difference of 0.19 is
more than 11 standard errors from zero. Value firms also are more likely to disinvest. For example,
the frequency of disinvestment is 0.59 for value firms, which is about 36% higher than that of growth
firms, 0.38. The disinvestment dummy, which effectively is a firm-level measure of the disinvestment
frequency, yields similar results: The average value of the dummy goes up from 0.18 for the growth
decile to 0.79 for the value decile. And the difference is more than seven standard errors from zero.
Consistent with prior studies such as Smith and Watts (1992), financial leverage and book-tomarket equity have a monotonically increasing relation: The leverage of growth firms is on average
0.32, whereas the leverage of value firms is on average 4.71. And the difference is more than 14
standard errors from zero. Table 5 shows that the operating leverage increases almost monotonically
from 1.74 for growth firms to 4.38 for value firms. The economic interpretation is that for one
percentage change in sales, we should expect to see on average 1.74% change in earnings for growth
firms but on average 4.38% earnings change for value firms. And the difference of 2.58% is almost
22
18 standard errors from zero. Finally, the composite inflexibility index also goes up monotonically
from 0.62 for growth firms to 2.78 for value firms, and the difference is again highly significant.
Does Inflexibility Increase the Cost of Equity?
After documenting significant differences in real flexibility between value and growth firms, we now
ask whether inflexibility increases the costs of equity capital of firms. The answer is affirmative.
We proceed in two steps. In the first step, we use the Fama-MacBeth (1973) two-step procedure:
Each month we regress monthly stock returns on the lagged flexibility measures, and then average
the time series of the regression coefficients and compute their t-statistics using the Newey-West
(1987) standard errors. Table 6 reports the regression results. The first two specifications show
that book-to-market is positively related to future returns with a slope of 0.38 (t = 6.08), and that
although the slope of the market equity is significant (t = −2.10), it is small in magnitude, −0.04.
More important, the subsequent five specifications report univariate cross-sectional regressions
on each of the four individual measures and the composite inflexibility index. Each of the inflexibility measures carries a positive coefficient, meaning that firms with less flexibility have higher costs
of equity. Among the four individual measures, financial leverage and operating leverage have positive slopes that are significant at the one percent confidence level (t = 4.06 and 3.05, respectively).
The composite index of inflexibility has a significant positive coefficient of 0.12, which is more than
three standard errors from zero. The last specification shows that the composite inflexibility index
remains significant with a coefficient of 0.07 (t = 1.98) after we control for book-to-market and size.
Size is no longer significant but book-to-market retains its strong explanatory power for average
returns with the presence of the composite inflexibility index.
In Table 7, we quantify the relation between inflexibility and costs of equity using the FamaFrench (1993) portfolio approach. We sort all stocks on their composite inflexibility index into
ten deciles at the end of June in each year t from 1972 to 2007. The equal-weighted and valueweighted returns for each portfolio are calculated from July of year t to June of year t + 1. From
23
Panel A, moving from the decile with the lowest inflexibility index to the decile with the highest
inflexibility index, the average equal-weighted returns increase from 0.87% to 1.90% per month.
The increasing relation is virtually monotonic. The high-minus-low inflexibility portfolio earns an
average equal-weighted return of 1.03% per month (t = 4.11). Controlling for the market beta and
for the Fama-French factor loadings does not eliminate the positive relation between inflexibility
and average equal-weighted returns. Moving from the low inflexibility to the high inflexibility decile,
the CAPM alpha increases from −0.34% to 0.94% per month and the Fama-French alpha rises from
−0.18% to 0.43% per month. Both differences are more than 3.7 standard errors from zero.
Panel B of Table 7 shows the value-weighted results. The basic result is unchanged: Firms with
less flexibility continue to earn higher average returns than firms with more flexibility. The spread
in the average value-weighted returns between the extreme deciles is 0.72% per month (t = 2.65).
Adjusting for the market beta does not affect the result: The spread in the CAPM alpha across the
two deciles is 0.86%, and is more than three standard errors from zero. When we use the Fama and
French (1993) three-factor model to explain the ten value-weighted deciles formed on inflexibility,
the alphas are all close to zero. And the difference between the low and high deciles is insignificant.
In sum, our results suggest a strong positive relation between real inflexibility and the costs of equity.
6
Conclusion
Using the two-state Markov switching framework of Perez-Quiros and Timmermann (2000), we
document new evidence that the expected value premium displays strong countercyclical variations. In recessions the expected excess returns of value stocks are most strongly affected, and the
expected excess returns of growth stocks are least affected, by worsening economic conditions as
measured by higher one-month Treasury bill rates and higher default spreads. By way of contrast,
in expansions the expected excess returns of both value and growth stocks have mostly insignificant
loadings on the two aggregate economic indicators. Because of these asymmetries across the states
of the economy, the expected value premium tends to spike upward rapidly during recessions only
24
to decline more gradually in the subsequent expansions. Our evidence suggests that conditioning
information is important in the cross section of average returns, lending support to the conditional
asset pricing literature (e.g, Ferson and Harvey 1991, 1993, 1999, Jagannathan and Wang 1996,
Campbell and Cochrane 1999, 2000, Lettau and Ludvigson 2001, and Ang and Chen 2001).
We also examine the potential sources of the time-variations in the expected value premium.
Using a variety of flexibility proxies such as the ratio of fixed assets to total assets, the frequency of
disinvestment, financial leverage, and operating leverage, we report new evidence that value firms
are less flexible in adjusting to worsening economic conditions than growth firms. Moreover, using a
composite inflexibility index that aggregates the information contained in the individual proxies, we
show that inflexibility increases the costs of equity in the cross section. Our evidence is largely consistent with the theoretical predictions from the emerging investment-based asset pricing literature
(e.g., Carlson, Fisher, and Giammarino 2004, Zhang 2005, and Livdan, Sapriza, and Zhang 2008).
25
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29
Table 1 : Moments of Monthly Excess Returns for Ten Book-to-Market Deciles (January
1954 to December 2007)
This table reports the mean, volatility, skewness, and kurtosis of the continuously compounded excess returns on the
decile portfolios formed on book-to-market equity. Excess returns are calculated as the difference between monthly
stock returns and the one-month Treasury bill rate. The data for the book-to-market deciles and the one-month
Treasury bill rate are from Kenneth French’s Web site.
Portfolio
Mean
Volatility
Skewness
Kurtosis
Growth
2
3
4
5
6
7
8
9
Value
0.482
0.546
0.615
0.610
0.680
0.722
0.721
0.858
0.892
0.966
5.037
4.548
4.513
4.431
4.226
4.210
4.198
4.294
4.529
5.235
−0.238
−0.435
−0.505
−0.424
−0.412
−0.369
0.027
−0.021
−0.103
0.023
4.408
4.942
5.559
5.342
5.907
5.511
4.896
5.118
5.037
6.160
Value-minus-growth
0.484
4.275
0.438
4.412
30
Table 2 : Parameter Estimates for the Univariate Markov Switching Model of Excess Returns
on Decile Portfolios Formed on Book-to-Market Equity (January 1954 to December 2007)
For each book-to-market decile portfolio, we estimate the following two-state Markov switching model:
rt
=
β 0,St + β 1,St T Bt−1 + β 2,St DEFt−1 + β 3,St ∆Mt−2 + β 4,St DIVt−1 + ǫt
ǫt
∼
N (0, σ2St ),
pt
=
P (St = 1|St−1 = 1) = Φ(π 0 + π 1 T Bt−1 )
St = {1, 2}
1 − pt
=
P (St = 2|St−1 = 1)
qt
=
P (St = 2|St−1 = 2) = Φ(π 0 + π 2 T Bt−1 )
1 − qt
=
P (St = 1|St−1 = 2)
in which rt is the monthly excess return for a given decile portfolio and St is the regime indicator. T B is the onemonth Treasury bill rate, DEF is the yield spread between Baa- and Aaa-rated corporate bonds, ∆M is the annual
rate of growth of the monetary base, and DIV is the dividend yield of the CRSP value-weighted portfolio. Standard
errors for the parameter estimates are reported in parentheses.
Growth
Constant, state 1
Constant, state 2
T B, state 1
T B, state 2
DEF , state 1
DEF , state 2
∆M , state 1
∆M , state 2
DIV , state 1
DIV , state 2
Transition probability
Constant
T B, state 1
T B, state 2
Standard deviation
σ, state 1
σ, state 2
Log likelihood value
−0.027 (0.01)
−0.019 (0.01)
−5.681 (1.54)
−1.454 (1.51)
4.019 (0.88)
1.769 (0.96)
0.077 (0.06)
−0.049 (0.06)
0.220 (0.34)
0.832 (0.28)
parameters
1.568 (0.40)
0.828 (0.92)
0.131 (1.03)
0.059
0.028
1056
(0.00)
(0.00)
Decile 6
Constant, state 1
Constant, state 2
T B, state 1
T B, state 2
DEF , state 1
DEF , state 2
∆M , state 1
∆M , state 2
DIV , state 1
DIV , state 2
Transition probability
Constant
T B, state 1
T B, state 2
Standard deviation
σ, state 1
σ, state 2
Log likelihood value
−0.060 (0.03)
0.009 (0.01)
−7.725 (3.92)
−2.309 (1.05)
5.951 (1.93)
0.392 (0.63)
0.095 (0.17)
0.000 (0.03)
0.697 (0.89)
0.181 (0.19)
parameters
1.704 (0.44)
−0.754 (0.82)
0.125 (0.73)
0.059
0.033
1165
(0.00)
(0.00)
Decile 2
Decile 3
Decile 4
Decile 5
−0.017
−0.009
−5.624
−1.406
3.848
1.763
0.045
−0.045
0.122
0.397
(0.01)
(0.01)
(1.63)
(1.91)
(0.83)
(1.07)
(0.06)
(0.06)
(0.37)
(0.26)
−0.015
−0.005
−6.439
−1.640
3.136
1.162
0.019
−0.031
0.486
0.473
(0.01)
(0.02)
(1.21)
(1.95)
(0.69)
(1.12)
(0.04)
(0.05)
(0.26)
(0.28)
−0.043
0.004
−6.980
−1.917
5.617
1.328
0.063
−0.011
0.271
0.055
(0.02)
(0.01)
(2.53)
(1.17)
(1.28)
(0.73)
(0.11)
(0.04)
(0.65)
(0.20)
−0.088
0.016
−7.185
−3.197
5.136
0.450
0.124
−0.029
1.795
0.112
(0.02)
(0.01)
(3.16)
(1.12)
(1.79)
(0.65)
(0.10)
(0.03)
(0.75)
(0.18)
1.647
0.889
0.155
(0.54)
(1.31)
(1.48)
1.994
0.783
−0.819
(0.57)
(1.71)
(1.92)
1.775
−0.047
0.128
(0.47)
(0.79)
(0.88)
2.178
−1.684
−1.124
(0.44)
(0.83)
(0.82)
0.053
0.030
1112
(0.00)
(0.00)
0.050
0.024
1127
(0.00)
(0.00)
0.059
0.033
1133
(0.00)
(0.00)
0.057
0.032
1170
(0.00)
(0.00)
Decile 7
Decile 8
Decile 9
Value
−0.088
0.021
−8.432
−1.983
6.180
−0.077
0.135
−0.069
1.605
0.085
(0.02)
(0.01)
(2.80)
(1.08)
(1.48)
(0.65)
(0.10)
(0.03)
(0.63)
(0.19)
−0.091
0.011
−8.810
−2.447
6.402
0.526
0.052
−0.01
1.839
0.223
(0.03)
(0.01)
(3.12)
(1.02)
(1.84)
(0.57)
(0.11)
(0.03)
(0.64)
(0.20)
−0.070
0.010
−9.350
−1.389
6.853
0.335
0.066
−0.019
1.190
0.196
(0.03)
(0.01)
(3.24)
(1.05)
(1.92)
(0.57)
(0.09)
(0.03)
(0.67)
(0.19)
−0.082
0.010
−11.667
−1.339
7.309
0.382
0.182
0.019
1.523
0.137
(0.03)
(0.01)
(3.28)
(1.12)
(1.79)
(0.62)
(0.12)
(0.03)
(0.75)
(0.22)
1.436
−0.869
−0.008
(0.22)
(0.41)
(0.27)
1.354
−0.806
0.439
(0.45)
(0.83)
(0.75)
1.278
−0.573
0.350
(0.40)
(0.80)
(0.76)
1.409
−0.056
0.752
(0.36)
(0.71)
(0.75)
0.054
0.031
1160
(0.00)
(0.00)
0.058
0.033
1150
(0.00)
(0.00)
0.062
0.031
1121
(0.00)
(0.00)
0.072
0.037
1035
(0.00)
(0.00)
31
Table 3 : Tests for Identical Slope Coefficients Across States in the Markov Switching Model (January 1954 to December 2007)
For each book-to-market decile, we estimate the following two-state Markov switching model:
rt
=
β 0,St + β 1,St T Bt−1 + β 2,St DEFt−1 + β 3,St ∆Mt−2 + β 4,St DIVt−1 + ǫt
ǫt
∼
N (0, σ2St ),
pt
=
P (St = 1|St−1 = 1) = Φ(π 0 + π 1 T Bt−1 )
St = {1, 2}
1 − pt
=
P (St = 2|St−1 = 1)
qt
=
P (St = 2|St−1 = 2) = Φ(π 0 + π 2 T Bt−1 )
1 − qt
=
P (St = 1|St−1 = 2)
in which rt is the monthly excess return for a given decile and St is the regime indicator. T B is the one-month Treasury bill rate, DEF is the yield spread between
32
Baa- and Aaa-rated corporate bonds, ∆M is the annual growth rate of the money supply, and DIV is the dividend yield of the CRSP value-weighted portfolio.
We conduct likelihood ratio tests on the null hypothesis that the coefficients are equalized across states, i.e., β k,St =1 = β k,St =2 , k = {1, 2, 3, 4}. The p-value is the
probability that the null hypothesis is not rejected. When testing the null hypothesis, we condition on the existence of two states in the conditional volatility.
Unrestricted log likelihood value
Restricted log likelihood with β k,St =1 = β k,St =2 , k = {1, 2, 3, 4}
p-value
Unrestricted log likelihood value
Restricted log likelihood with β k,St =1 = β k,St =2 , k = {1, 2, 3, 4}
p-value
Growth
Decile 2
Decile 3
Decile 4
Decile 5
1056
1047
0.00
1112
1105
0.01
1127
1120
0.01
1133
1130
0.33
1170
1162
0.00
Decile 6
Decile 7
Decile 8
Decile 9
Value
1165
1162
0.14
1160
1138
0.00
1150
1148
0.41
1121
1102
0.00
1034
1014
0.00
Table 4 : The Joint Markov Switching Model for Excess Returns on Book-to-Market Decile
Ten (Value) and Decile One (Growth) Portfolios (January 1954 to December 2007)
We estimate the following joint Markov switching model for excess returns on book-to-market deciles ten (value)
and one (growth):
rt
=
β 0,St + β 1,St T Bt−1 + β 2,St DEFt−1 + β 3,St ∆Mt−2 + β 4,St DIVt−1 + ǫt
ǫt
∼
N (0, ΣSt ),
log(Σii,St )
=
λiSt
Σij,St
=
ρSt (Σii,St )1/2 (Σjj,St )1/2 ,
pt
=
P (St = 1|St−1 = 1) = Φ(π 0 + π 1 T Bt−1 )
St = {1, 2}
i 6= j
1 − pt
=
P (St = 2|St−1 = 1)
qt
=
P (St = 2|St−1 = 2) = Φ(π 0 + π 2 T Bt−1 )
1 − qt
=
P (St = 1|St−1 = 2)
in which rt = (rtg , rtv )′ is the (2×1) vector that contains the monthly excess returns on the growth and value portfolios,
rtg and rtv , respectively. β k,St , k = 0, 1, 2, 3, 4, is a (2 × 1) vector with elements β k,St = (β gk,St , β vk,St )′ . ǫt ∼ N (0, ΣSt )
is a vector of residuals. ΣSt is a positive semidefinite (2 × 2) matrix containing the variances and covariances of the
residuals of the value and growth portfolio excess returns in state St . The diagonal elements of this variance-covariance
matrix, Σii,St , take the similar form as in the univariate model: log(Σii,St ) = λiSt . The off-diagonal terms, Σij,St ,
assume a state-dependent correlation between the residuals, denoted ρSt , i.e., Σij,St = ρSt (Σii,St )1/2 (Σjj,St )1/2 for
i 6= j. T B is the one-month Treasury bill rate, DEF is the yield spread between Baa- and Aaa-rated corporate
bonds, ∆M is the annual rate of growth of the monetary base, and DIV is the dividend yield of the value-weighted
market portfolio. Standard errors are in parentheses to the right of the parameter estimates. The correlation and
transition probabilities are common to both deciles. For each set of coefficients, the p-value from the likelihood ratio
tests reports the probability of the restriction that the asymmetry between the value and growth portfolios is identical
against the alternative that the asymmetry is larger for the value portfolio.
Growth (g)
Value (v)
Tests for identical asymmetries
Constant, state 1
Constant, state 2
−0.043
−0.004
(0.02)
(0.01)
−0.047
−0.003
(0.02)
(0.01)
Constant: β g0,1 − β g0,2 = β v0,1 − β v0,2
Log likelihood value
p-value
T B, state 1
T B, state 2
−6.741
−1.725
(2.18)
(1.47)
−10.758
−2.965
(2.25)
(1.41)
T B: β g1,1 − β g1,2 = β v1,1 − β v1,2
Log likelihood value
p-value
2241
(0.02)
(1.29)
(0.77)
DEF : β g2,1 − β g2,2 = β v2,1 − β v2,2
Log likelihood value
p-value
2240
(0.01)
(0.08)
(0.06)
∆M : β g3,1 − β g3,2 = β v3,1 − β v3,2
Log likelihood value
p-value
2240
(0.01)
DIV : β g4,1 − β g4,2 = β v4,1 − β v4,2
Log likelihood value
p-value
2241
(0.02)
DEF , state 1
DEF , state 2
∆M , state 1
∆M , state 2
4.599
1.057
0.093
0.024
DIV , state 1
0.448
DIV , state 2
0.420
Standard deviation parameters
σ, state 1
0.064
σ, state 2
0.037
(1.38)
(0.72)
(0.08)
(0.06)
7.761
1.285
0.081
0.084
(0.49)
(0.47)
0.387
0.430
(0.25)
(0.25)
(0.00)
(0.00)
0.070
0.037
(0.00)
(0.00)
2243
(0.30)
Parameters common to both deciles
Correlation parameters
ρ, state 1
ρ, state 2
Transition probability parameters
Constant
T B, state 1
T B, state 2
Unconstrained log likelihood
0.638
0.665
(0.05)
(0.04)
1.284
0.250
0.375
2244
(0.34)
(0.58)
(0.68)
33
T B: π 1 = π2
Log likelihood value
p-value
2242
(0.06)
Table 5 : Measures of Flexibility Across the Book-to-Market Deciles (1972 to 2007)
At the end of June of each year t, stocks are allocated into deciles based on book-to-market equity in fiscal year ending
in calendar year t − 1. We report the summary statistics of the flexibility proxies in the year prior to the portfolio
formation date. Book-to-market equity, B/M , is measured as in Davis, Fama, and French (2000) where book equity
is the stockholders book equity (Compustat annual item 60), plus balance sheet deferred taxes and investment tax
credit (item 35), minus book value of preferred stock (in the following order: item 56 or item 10 or item 130). The
market equity is the price times shares outstanding at the end of December of calendar year t − 1. Fixed to total
assets ratio, F A/T A, is the ratio of fixed assets (item 7) to total assets (item 6). Disinvestment frequency is the
ratio of the number of firms that have disinvested (item 107 is positive) in the past three years divided by the total
number of firms in the portfolio. Disinvestment dummy is an indicative variable that takes a value of one if item 107
(sale of property, plant and equipment) is positive for at least one year in the past three years and takes a value of
zero otherwise. Financial leverage, A/M E, is the ratio of book assets (item 6) to market value of equity. Operating
leverage is the ratio of the percentage change in operating income before depreciation (item 13) to the percentage
change in sales (item 12). The table entries are the time series averages of annual cross-sectional medians, except
for the disinvestment frequency, which is the time series average of annual disinvestment frequency of a given decile.
The heteroscedasticity-and-autocorrelation-consistent t-statistics that test the differences in characteristics between
the value and growth portfolios are individually zero also are reported.
Decile
B/M
F A/T A
Disinvestment
frequency
Disinvestment
dummy
A/M E
Operating
leverage
Composite
inflexibility
Growth
2
3
4
5
6
7
8
9
Value
Value-minus-growth
t(value-minus-growth)
0.15
0.29
0.42
0.55
0.69
0.84
1.01
1.25
1.61
2.53
2.38
14.85
0.36
0.38
0.41
0.44
0.46
0.48
0.49
0.50
0.52
0.55
0.19
11.82
0.38
0.43
0.47
0.50
0.51
0.54
0.55
0.56
0.59
0.59
0.21
17.51
0.18
0.24
0.40
0.53
0.59
0.59
0.59
0.68
0.74
0.79
0.62
7.41
0.32
0.55
0.78
1.02
1.29
1.56
1.88
2.29
2.97
4.71
4.40
14.12
1.74
1.64
1.74
1.86
2.01
2.17
2.41
2.73
3.21
4.38
2.64
17.93
3.48
4.01
4.54
4.95
5.33
5.69
5.98
6.30
6.73
7.11
3.63
38.16
34
Table 6 : Fama-MacBeth Cross-Sectional Regressions of Monthly Excess Returns on
Measures of Flexibility (January 1972 to December 2007)
Fixed to total assets, F A/T A, is the ratio of fixed assets (Compustat annual item 7) to total assets (item 6). Financial
leverage, A/M E, is the ratio of book assets (item 6) to market value of equity. Operating leverage is the ratio of
the percentage change in operating income before depreciation (item 13) to the percentage change in sales (item 12).
The independent variables, F A/T A, A/M E, and operating leverage are integers from one to ten that represent the
sorted decile index of the corresponding variable with ten being a highest value. The disinvestment dummy is an
indicator variable that takes a value of one if a firm has disinvested in at least one year during the past three years
over fiscal years ending calendar years t − 4 to t − 1 and takes a value of zero otherwise. The composite flexibility
index is a sorted decile index of an arithmetic average of decile indexes of the four flexibility proxies. Higher value of
this the flexibility index means less flexibility. Control variables include book-to-market equity, B/M , measured in
the fiscal year ending in calendar year t − 1, and size, measured as the market value of equity at the June of year t.
Estimates of the regression coefficients are the time series averages of monthly cross-sectional regression coefficients.
The t-statistics, in parenthesis, are adjusted for autocorrelations.
Regression
B/M
1
0.38
(6.05)
Size
F A/T A
Disinvestment
dummy
Operating
leverage
Composite
inflexibility
−0.04
(−2.10)
2
3
0.04
(1.41)
4
0.02
(1.73)
5
0.12
(4.06)
6
0.05
(3.05)
7
8
A/M E
0.33
(6.12)
0.12
(3.17)
0.07
(1.98)
−0.03
(−1.62)
35
Table 7 : Decile Portfolios Formed on the Composite Flexibility Index (January 1972 to December 2007)
At the end of June of each year t, we sort stocks into deciles based on the composite flexibility index as of fiscal year ending in calendar year t − 1. The composite
flexibility index is defined as the arithmetic average of decile indexes of four flexibility proxies including the ratio of fixed assets to total assets, financial leverage,
operating leverage, and disinvestment frequency. Equal- and value-weighted portfolio returns are calculated from July of year t to June of year t + 1, and the
portfolios are rebalanced in each June. For each composite flexibility decile and the high-minus-low decile, we report average monthly returns, CAPM alpha, the
alpha from the Fama-French (1993) three-factor model, and the Sharpe Ratio. The t-statistics that test the average return, CAPM alpha, and Fama-French alpha
equal zero are reported in parentheses. The t-statistics are adjusted for heteroscedasticity and autocorrelations.
Deciles formed on the composite flexibility index
Low
2
3
4
5
6
36
7
8
9
High
H−L
tH−L
1.45
(5.12)
0.43
(2.49)
0.12
(1.19)
1.52
(5.29)
0.50
(2.77)
0.13
(1.29)
1.65
(5.79)
0.65
(3.54)
0.18
(1.76)
1.90
(6.37)
0.94
(4.73)
0.43
(3.30)
1.03
(4.11)
1.28
(5.74)
0.60
(3.76)
0.99
(3.99)
0.00
(0.01)
−0.05
(−0.45)
1.14
(5.23)
0.18
(1.65)
0.02
(0.22)
1.33
(6.30)
0.32
(2.48)
0.02
(0.21)
1.41
(5.77)
0.40
(2.54)
−0.10
(−0.80)
0.72
(2.65)
0.86
(3.23)
−0.13
(−0.68)
Panel A: Equal-weighted portfolios
Average return
CAPM alpha
Fama-French alpha
0.87
(1.90)
−0.34
(−1.23)
−0.18
(−0.95)
1.20
(3.52)
0.09
(0.43)
0.06
(0.43)
1.26
(3.95)
0.19
(1.00)
0.03
(0.30)
1.28
(4.24)
0.22
(1.16)
0.05
(0.40)
1.42
(4.77)
0.36
(2.04)
0.16
(1.45)
1.46
(4.98)
0.41
(2.34)
0.19
(1.79)
Panel B: Value-weighted portfolios
Average return
CAPM alpha
Fama-French alpha
0.69
(1.88)
−0.46
(−2.48)
0.02
(0.15)
0.94
(3.59)
−0.07
(−0.56)
0.16
(1.44)
0.95
(3.20)
−0.11
(−1.28)
−0.08
(−0.90)
0.87
(3.89)
−0.12
(−1.30)
−0.08
(−0.82)
0.95
(4.14)
−0.02
(−0.29)
−0.01
(−0.17)
1.22
(5.45)
0.23
(2.60)
0.14
(1.60)
Figure 1 : Year-by-Year Returns of the Value-Minus-Growth Strategy (Book-to-Market
Decile Ten Minus Decile One), 1954 to 2007
This figure presents the annual buy-and-hold returns for the book-to-market decile ten minus decile one. The annual
returns for the book-to-market deciles are from Kenneth French’s Web site.
0.6
0.5
0.4
0.3
0.2
0.1
0
−0.1
−0.2
−0.3
−0.4
1950
1960
1970
1980
37
1990
2000
2010
Figure 2 : Univariate Markov Switching Model, Probability of High Variance and Low Mean
State (January 1954 to December 2007)
Time series of the probability of being in state 1 at time t conditional on information in period t − 1 is plotted
for the value portfolio (Panel A) and for the growth portfolio (Panel B). We identify value and growth using the
deciles formed on book-to-market equity available from Kenneth French’s Web site. The value portfolio is defined as
the decile with the highest book-to-market equity and the growth portfolio is defined as the decile with the lowest
book-to-market equity. Shaded areas indicate NBER recession periods.
Panel A: Decile 10 (Value)
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
1954.1
1961.1
1969.6
1977.3
1984.11
1992.8
2000.4
2007.12
1992.8
2000.4
2007.12
Panel B: Decile 1 (Growth)
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
1954.1
38
1961.1
1969.6
1977.3
1984.11
Figure 3 : Expected Excess Returns, Univariate and Bivariate Markov Switching Models
This figure plots the expected excess returns for the value portfolio (Panel A), the growth portfolio (Panel B), and
their difference (Panel C) from the univariate and bivariate Markow switching models in Tables 2 and 4. The solid
lines use parameter estimates in the bivariate Markov switching model, and the dashed lines use the estimates from
the univariate model. We identify value and growth using the deciles formed on book-to-market equity available from
Kenneth French’s Web site. The value portfolio is the decile with the highest book-to-market equity and the growth
portfolio is the decile with the lowest book-to-market equity. Shaded areas indicate NBER recession periods.
Panel A: Decile 10 (Value)
0.1
0.05
0
−0.05
Univariate
Bivariate
−0.1
1954.1
1961.1
1969.6
1977.3
1984.11
1992.8
2000.4
2007.12
Panel B: Decile 1 (Growth)
0.06
0.04
0.02
0
−0.02
−0.04
Univariate
Bivariate
−0.06
1954.1
1961.1
1969.6
1977.3
1984.11
1992.8
2000.4
2007.12
Panel C: Decile 10 − Decile 1
0.04
0.02
0
−0.02
39
−0.04
1954.1
Univariate
Bivariate
1961.1
1969.6
1977.3
1984.11
1992.8
2000.4
2007.12
Figure 4 : Bivariate Markov Switching Model, Conditional Volatilities and Conditional
Sharpe Ratios (January 1954 to December 2007)
Panel A plots the standard deviation of the conditional returns for the value and growth portfolios. Panel B plots
the conditional Sharpe ratio defined as the expected excess return over the conditional volatility. The solid lines are
for value firms and the dotted lines are for growth firms. We identify value and growth using the deciles formed on
book-to-market equity available from Kenneth French’s Web site. The value portfolio is defined as the decile with
the highest book-to-market equity and the growth portfolio is defined as the decile with the lowest book-to-market
equity. Shaded areas indicate NBER recession periods.
Panel A: Conditional Volatilities
0.075
0.07
0.065
0.06
0.055
0.05
0.045
0.04
Value
Growth
0.035
0.03
1954.1
1961.1
1969.6
1977.3
1984.11
1992.8
2000.4
2007.12
Panel B: Conditional Sharpe Ratios
1.5
1
0.5
0
−0.5
−1
Value
Growth
40
−1.5
1954.1
1961.1
1969.6
1977.3
1984.11
1992.8
2000.4
2007.12
Figure 5 : Expected Returns, Conditional Volatility, and Conditional Sharpe Ratio for the
Value-minus-Growth Portfolio Estimated from the Univariate Markov Switching Model
For the value-minus-growth portfolio, this figure plots the expected return (Panel A), the conditional volatility (Panel
B), and the conditional Sharpe ratio (Panel C) from the univariate Markow switching model. We identify value and
growth using the deciles formed on book-to-market equity available from Kenneth French’s Web site. The value
portfolio is the decile with the highest book-to-market equity and the growth portfolio is the decile with the lowest
book-to-market equity. Shaded areas indicate NBER recession periods.
Panel A: Expected Return
0.05
0.04
0.03
0.02
0.01
0
−0.01
1954.1
1961.1
1969.6
1977.3
1984.11
1992.8
2000.4
2007.12
1992.8
2000.4
2007.12
2000.4
2007.12
Panel B: Conditional Volatility
0.065
0.06
0.055
0.05
0.045
0.04
1954.1
1961.1
1969.6
1977.3
1984.11
Panel C: Conditional Sharpe Ratio
0.6
0.4
0.2
0
−0.2
1954.1
41
1961.1
1969.6
1977.3
1984.11
1992.8